By Adalbert Kerber

The subject of this booklet is finite team activities and their use so that it will strategy finite unlabeled buildings by means of defining them as orbits of finite teams of units. famous examples are graphs, linear codes, chemical isomers, spin configurations, isomorphism periods of combinatorial designs etc.

This moment variation is a longer model and places extra emphasis on functions to the confident conception of finite buildings. fresh development during this box, particularly in layout and coding thought, is described.

This booklet may be of significant use to researchers and graduate students.

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**Sample text**

Now we assume two species R and S where S[0] = 0. The partitional composition of and is defined by n s R(S)[M] := u R[p] x (xiS[pil). p={PO·····Pr-1 }EPar[M] This means that in order to obtain R(S)[M] we first of all form the set partitions p of M. ) blocks Pi, i = 0, ... , r - 1, we form on each of these blocks Pi the set of S-structures, i. e. we form the sets S[pi ], and then the cartesian product over i = 0, ... , r- 1 of all these sets of S-structures together with R[p] (the set of R-structures on the set p which is of order r ).

8 Application (isometry classes of linear codes) It was mentioned that the isometry group on GF(q)n (with respect to the Hamming metric) is the wreath product G F (q )* 1 Sn. It also acts on the power set of G F(q )n, which is yX := 2(GF(q)"). The set of orbits of the isometry group on this usually very big power set contains particular orbits that consist of subspaces only. These orbits are called the isometry classes of linear codes of length n. Hence the isometry classes of linear codes of length n over G F(q) are elements of the set of orbits GF(q)* I Sn ~2(GF(q)").

The corresponding permutation group will be denoted by G ® H and called the cartesian product of G and ii. 15 Application (Mackey's Theorem) Assume two finite and transitive actions of G on X and Y. They yield, as was just described, a canonical action of G x G on X x Y which has as one of its restrictions the action of 11 ( G x G), the diagonal, which is isomorphic to G, on X x Y. 6): - Each orbit of G on X x Y contains an element of the form (x, gy ). to the action of GonG j(Gx n gGyg- 1). 1(G x G) ~(X x Y)).